Chapter Test
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If-then Form: If two planes intersect at a line, then the planes are not parallel.
The Converse: If two planes are not parallel, then the planes intersect at a line.
The Inverse: If two planes do not intersect at a line, then the planes are parallel.
The Contrapositive: If two planes do not intersect at a line, then the planes are parallel.
The Biconditional: Two planes intersect at a line if and only if the planes are not parallel.
Let's start by writing the if-then form. What can we say about two planes that intersect? Well, for one, they will not be parallel. We can use this fact to write a possible if-then statement.
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If two planes intersect at a line, then the planes are not parallel. |
Therefore, our hypothesis is that two planes intersect at a line, and our conclusion is that they cannot be parallel.
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If two planes are not parallel, then the planes intersect at a line. |
To write the inverse of a conditional statement, we negate both the hypothesis and the conclusion. This means each of the statements should state the opposite.
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If two planes do not intersect at a line, then the planes are parallel. |
To write the contrapositive of a conditional statement, first recall the converse we wrote above.
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If two planes are not parallel, then the planes intersect at a line. |
We negate the hypothesis and conclusion of the converse to write the contrapositive. Note that negating a negative means removing the not.
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If two planes are parallel, then the planes do not intersect at a line. |
If a conditional statement and its converse are both true, we can write it as a biconditional statement which is a statement containing the phrase if and only if.
Algebraically it is written with a special symbol.
hypothesis⟺ conclusion
Since both the conditional statement and its converse applies, we can write it as the following biconditional statement.
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Two planes intersect at a line if and only if the planes are not parallel. |